Simplify -((abc(a^-2c^-3)^-3)/(c^7))^-7a^49b^7c^21+1

$- {(\frac{a b c {(a^{- 2} c^{- 3})}^{- 3}}{c^{7}})}^{- 7} a^{49} b^{7} c^{21} + 1$

Simplify each term.

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Move ${(a^{- 2} c^{- 3})}^{- 3}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$- {(\frac{a b c}{c^{7} {(a^{- 2} c^{- 3})}^{3}})}^{- 7} a^{49} b^{7} c^{21} + 1$

Cancel the common factor of $c$ and $c^{7}$ .

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Factor $c$ out of $a b c$ .

$- {(\frac{c (a b)}{c^{7} {(a^{- 2} c^{- 3})}^{3}})}^{- 7} a^{49} b^{7} c^{21} + 1$

Cancel the common factors.

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Factor $c$ out of $c^{7} {(a^{- 2} c^{- 3})}^{3}$ .

$- {(\frac{c (a b)}{c (c^{6} {(a^{- 2} c^{- 3})}^{3})})}^{- 7} a^{49} b^{7} c^{21} + 1$

Cancel the common factor.

$- {(\frac{c (a b)}{c (c^{6} {(a^{- 2} c^{- 3})}^{3})})}^{- 7} a^{49} b^{7} c^{21} + 1$

Rewrite the expression.

$- {(\frac{a b}{c^{6} {(a^{- 2} c^{- 3})}^{3}})}^{- 7} a^{49} b^{7} c^{21} + 1$

Simplify the denominator.

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Apply the product rule to $a^{- 2} c^{- 3}$ .

$- {(\frac{a b}{c^{6} {(a^{- 2})}^{3} {(c^{- 3})}^{3}})}^{- 7} a^{49} b^{7} c^{21} + 1$

Multiply the exponents in ${(a^{- 2})}^{3}$ .

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Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$- {(\frac{a b}{c^{6} a^{- 2 \cdot 3} {(c^{- 3})}^{3}})}^{- 7} a^{49} b^{7} c^{21} + 1$

Multiply $- 2$ by $3$ .

$- {(\frac{a b}{c^{6} a^{- 6} {(c^{- 3})}^{3}})}^{- 7} a^{49} b^{7} c^{21} + 1$

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$- {(\frac{a b}{c^{6} \frac{1}{a^{6}} {(c^{- 3})}^{3}})}^{- 7} a^{49} b^{7} c^{21} + 1$

Multiply the exponents in ${(c^{- 3})}^{3}$ .

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Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$- {(\frac{a b}{c^{6} \frac{1}{a^{6}} c^{- 3 \cdot 3}})}^{- 7} a^{49} b^{7} c^{21} + 1$

Multiply $- 3$ by $3$ .

$- {(\frac{a b}{c^{6} \frac{1}{a^{6}} c^{- 9}})}^{- 7} a^{49} b^{7} c^{21} + 1$

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$- {(\frac{a b}{c^{6} \frac{1}{a^{6}} \frac{1}{c^{9}}})}^{- 7} a^{49} b^{7} c^{21} + 1$

Simplify the denominator.

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Combine $c^{6}$ and $\frac{1}{a^{6}}$ .

$- {(\frac{a b}{\frac{c^{6}}{a^{6}} \cdot \frac{1}{c^{9}}})}^{- 7} a^{49} b^{7} c^{21} + 1$

Multiply $\frac{c^{6}}{a^{6}}$ and $\frac{1}{c^{9}}$ .

$- {(\frac{a b}{\frac{c^{6}}{a^{6} c^{9}}})}^{- 7} a^{49} b^{7} c^{21} + 1$

Reduce the expression $\frac{c^{6}}{a^{6} c^{9}}$ by cancelling the common factors.

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Multiply by $1$ .

$- {(\frac{a b}{\frac{c^{6} \cdot 1}{a^{6} c^{9}}})}^{- 7} a^{49} b^{7} c^{21} + 1$

Factor $c^{6}$ out of $a^{6} c^{9}$ .

$- {(\frac{a b}{\frac{c^{6} \cdot 1}{c^{6} (a^{6} c^{3})}})}^{- 7} a^{49} b^{7} c^{21} + 1$

Cancel the common factor.

$- {(\frac{a b}{\frac{c^{6} \cdot 1}{c^{6} (a^{6} c^{3})}})}^{- 7} a^{49} b^{7} c^{21} + 1$

Rewrite the expression.

$- {(\frac{a b}{\frac{1}{a^{6} c^{3}}})}^{- 7} a^{49} b^{7} c^{21} + 1$

Multiply the numerator by the reciprocal of the denominator.

$- {(a b (a^{6} c^{3}))}^{- 7} a^{49} b^{7} c^{21} + 1$

Multiply $a$ by $a^{6}$ by adding the exponents.

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Move $a^{6}$ .

$- {(a^{6} a b c^{3})}^{- 7} a^{49} b^{7} c^{21} + 1$

Multiply $a^{6}$ by $a$ .

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Raise $a$ to the power of $1$ .

$- {(a^{6} a^{1} b c^{3})}^{- 7} a^{49} b^{7} c^{21} + 1$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- {(a^{6 + 1} b c^{3})}^{- 7} a^{49} b^{7} c^{21} + 1$

Add $6$ and $1$ .

$- {(a^{7} b c^{3})}^{- 7} a^{49} b^{7} c^{21} + 1$

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$- \frac{1}{{(a^{7} b c^{3})}^{7}} a^{49} b^{7} c^{21} + 1$

Simplify the denominator.

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Apply the product rule to $a^{7} b c^{3}$ .

$- \frac{1}{{(a^{7} b)}^{7} {(c^{3})}^{7}} a^{49} b^{7} c^{21} + 1$

Apply the product rule to $a^{7} b$ .

$- \frac{1}{{(a^{7})}^{7} b^{7} {(c^{3})}^{7}} a^{49} b^{7} c^{21} + 1$

Multiply the exponents in ${(a^{7})}^{7}$ .

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Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$- \frac{1}{a^{7 \cdot 7} b^{7} {(c^{3})}^{7}} a^{49} b^{7} c^{21} + 1$

Multiply $7$ by $7$ .

$- \frac{1}{a^{49} b^{7} {(c^{3})}^{7}} a^{49} b^{7} c^{21} + 1$

Multiply the exponents in ${(c^{3})}^{7}$ .

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Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$- \frac{1}{a^{49} b^{7} c^{3 \cdot 7}} a^{49} b^{7} c^{21} + 1$

Multiply $3$ by $7$ .

$- \frac{1}{a^{49} b^{7} c^{21}} a^{49} b^{7} c^{21} + 1$

Cancel the common factor of $a^{49}$ .

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Move the leading negative in $- \frac{1}{a^{49} b^{7} c^{21}}$ into the numerator.

$\frac{- 1}{a^{49} b^{7} c^{21}} a^{49} b^{7} c^{21} + 1$

Factor $a^{49}$ out of $a^{49} b^{7} c^{21}$ .

$\frac{- 1}{a^{49} (b^{7} c^{21})} a^{49} b^{7} c^{21} + 1$

Cancel the common factor.

$\frac{- 1}{a^{49} (b^{7} c^{21})} a^{49} b^{7} c^{21} + 1$

Rewrite the expression.

$\frac{- 1}{b^{7} c^{21}} b^{7} c^{21} + 1$

Cancel the common factor of $b^{7}$ .

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Factor $b^{7}$ out of $b^{7} c^{21}$ .

$\frac{- 1}{b^{7} (c^{21})} b^{7} c^{21} + 1$

Cancel the common factor.

$\frac{- 1}{b^{7} c^{21}} b^{7} c^{21} + 1$

Rewrite the expression.

$\frac{- 1}{c^{21}} c^{21} + 1$

Cancel the common factor of $c^{21}$ .

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Cancel the common factor.

$\frac{- 1}{c^{21}} c^{21} + 1$

Rewrite the expression.

$- 1 + 1$

Add $- 1$ and $1$ .

$0$

Simplify -((abc(a^-2c^-3)^-3)/(c^7))^-7a^49b^7c^21+1

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